This paper proposes a novel concept of exponential superiority in probability to compare the numerical methods for general stochastic differential equations from the perspective of the tail probability of the error. We take the linear stochastic oscillator as the test equation and consider several concrete numerical methods. By establishing the large deviation principles of the errors of the considered numerical methods, we show that the symplectic methods are exponentially superior to the non-symplectic methods in probability when the computational time $T$ is sufficiently large. This provides a new way to explain the superiority of stochastic symplectic methods over non-symplectic methods in the long-time simulation.

翻译：本文提出一个名为“概率指数优越性”的新概念，从误差尾部概率角度比较一般随机微分方程的数值方法。以线性随机振子为测试方程，考虑若干具体数值方法。通过建立所考察数值方法误差的大偏差原理，我们证明当计算时间$T$充分大时，辛方法在概率意义上指数优于非辛方法。这为解释随机辛方法在长时间模拟中相对非辛方法的优越性提供了新途径。